6 ms·

It's the smallest number bigger than 0.

by traderjane 6y ago

naringas 6y ago
strictly bigger than 0
- gspr 6y ago
  Doesn't exist.
TuringTest 6y ago
bigger or equal
- gspr 6y ago
  0.
thinkharderdev 6y ago
That doesn't exist. An open interval doesn't have a smallest number.
- da_chicken 6y ago
  Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal https://en.wikipedia.org/wiki/Infinitesimal If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly unique property for numbers.
  - pwdisswordfish2 6y ago
    There is no smallest positive infinitesimal either. At least in theories that manage to define those rigorously. And it’s mostly a formal trick anyway; standard epsilon-delta calculus avoids them entirely. Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?
    - da_chicken 6y ago
      That's a funny way to say, "No, I think you misunderstand. I mean to say no single infinitesimal number exists. Like infinity, the concept exists, but as a literal single number, no."
    - camelite 6y ago
      > There is no smallest positive infinitesimal either. There is in the surreal and hyperreal number systems. I got that from skimming wikipedia though....
  - gspr 6y ago
    > If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. Not at all. Standard calculus uses standard real numbers, for which there is no infinitesimal. One may well speak of infinitesimals as a mental tool when building a mental model for calculus, but those infinitesimals are not actual real numbers (or a well-defined mathematical object at all - in standard calculus).
    - brlewis 6y ago
      Correct. This is covered in the article. https://en.wikipedia.org/wiki/0.999...#Infinitesimals https://en.wikipedia.org/wiki/0.999...#Infinitesimals
  - petters 6y ago
    At the very least, don't write "Of course it does". It does not in the real number system.
  - deleted 6y ago
    [deleted]
- SkyBelow 6y ago
  It does exist. The other poster just clearly showed that it exists by referring to it. The problem is that if we include such a number in our formal system of math, we quickly find contradictions and the whole system falls apart. So such a number is incompatible with any formal system of math (though I guess you could start building one which does include such a number and see what properties it has). Herein lies the problem, the people you are talking with do not use a form system. There system of math has something similar to the same flaw of their system of grouping of things, which would include the whole grouping that contains every grouping that doesn't contain itself. People rarely deal in formal systems and thus they can handle completely illogical statements fine as long they are protected from seeing the consequence of it.
  - gowld 6y ago
    In math, when assuming the existence of something proved a contradiction, we conclude that the thing does not exist. The description may exist "integer between 3 and 4", but there is not described object. A description names a set or a class, and that class can have 0,1, or more numbers.
    - SkyBelow 6y ago
      >In math, when assuming the existence of something proved a contradiction, we conclude that the thing does not exist. Well only to the extent that you don't want to throw away any of the other axioms. Sometimes you do and there are some fun systems of math, but few have any practicality and those that do are often so advanced that even someone with an undergraduate focus in math can't appreciate those systems. It is much the same with computer science. I personally enjoyed playing around with formal concepts of computation and adding some extras to see what happens. For example, what happens to a Turing machine if part of the machine can time travel or has access to an oracle. Does this make concepts like time travel inherently contradictory to our notion of computation? But the practicality of these exercises does not exceed their entertainment value.
  - thinkharderdev 6y ago
    You are certainly correct that people arguing the opposite side probably don't have a formal system in mind, but I think the intuition that an open interval in the Reals doesn't have a smallest number is easy to grasp even without any formal training. So you can force them to see the consequences of it through fairly straightforward logical contradictions. Assume x is the smallest real number greater than 0. Then x/2 is also a real number and is greater than 0 but less than x. Therefore, x can't be the smallest real number greater than 0.
brlewis 6y ago
Why is 0.000... bigger than 0?
- gspr 6y ago
  It isn't.
  - brlewis 6y ago
    You are correct. This was a rhetorical question to get traderjane to question whether "bigger than 0" really applies here.
  - SkyBelow 6y ago
    Or is it? Say I'm a layman and I decide that in the system of math as I understand it, 0.000... is larger than 0. Yes, if I was going to be completely form with my own system of math I would eventually have to face the problems this introduces and resolve it, but until then I can generally adopt a self contradictory system and continue to live my life unaffected. Much like many people live their whole lives using naive set theory for their understanding of sets.
    - brlewis 6y ago
      Then in your system of math 0.999... is also less than 1. However, basic arithmetic taught to children requires that adding trailing zeros does not change the value of a number. You'll have a hard time doing arithmetic once you change that assumption.

It's the smallest number bigger than 0.

naringas 6y ago

strictly bigger than 0

gspr 6y ago
Doesn't exist.

TuringTest 6y ago

bigger or equal

gspr 6y ago
0.

thinkharderdev 6y ago

That doesn't exist. An open interval doesn't have a smallest number.

da_chicken 6y ago
Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal https://en.wikipedia.org/wiki/Infinitesimal If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly unique property for numbers.
- pwdisswordfish2 6y ago
  There is no smallest positive infinitesimal either. At least in theories that manage to define those rigorously. And it’s mostly a formal trick anyway; standard epsilon-delta calculus avoids them entirely. Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?
  - da_chicken 6y ago
    That's a funny way to say, "No, I think you misunderstand. I mean to say no single infinitesimal number exists. Like infinity, the concept exists, but as a literal single number, no."
  - camelite 6y ago
    > There is no smallest positive infinitesimal either. There is in the surreal and hyperreal number systems. I got that from skimming wikipedia though....
- gspr 6y ago
  > If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. Not at all. Standard calculus uses standard real numbers, for which there is no infinitesimal. One may well speak of infinitesimals as a mental tool when building a mental model for calculus, but those infinitesimals are not actual real numbers (or a well-defined mathematical object at all - in standard calculus).
  - brlewis 6y ago
    Correct. This is covered in the article. https://en.wikipedia.org/wiki/0.999...#Infinitesimals https://en.wikipedia.org/wiki/0.999...#Infinitesimals
- petters 6y ago
  At the very least, don't write "Of course it does". It does not in the real number system.
- deleted 6y ago
  [deleted]
SkyBelow 6y ago
It does exist. The other poster just clearly showed that it exists by referring to it. The problem is that if we include such a number in our formal system of math, we quickly find contradictions and the whole system falls apart. So such a number is incompatible with any formal system of math (though I guess you could start building one which does include such a number and see what properties it has). Herein lies the problem, the people you are talking with do not use a form system. There system of math has something similar to the same flaw of their system of grouping of things, which would include the whole grouping that contains every grouping that doesn't contain itself. People rarely deal in formal systems and thus they can handle completely illogical statements fine as long they are protected from seeing the consequence of it.
- gowld 6y ago
  In math, when assuming the existence of something proved a contradiction, we conclude that the thing does not exist. The description may exist "integer between 3 and 4", but there is not described object. A description names a set or a class, and that class can have 0,1, or more numbers.
  - SkyBelow 6y ago
    >In math, when assuming the existence of something proved a contradiction, we conclude that the thing does not exist. Well only to the extent that you don't want to throw away any of the other axioms. Sometimes you do and there are some fun systems of math, but few have any practicality and those that do are often so advanced that even someone with an undergraduate focus in math can't appreciate those systems. It is much the same with computer science. I personally enjoyed playing around with formal concepts of computation and adding some extras to see what happens. For example, what happens to a Turing machine if part of the machine can time travel or has access to an oracle. Does this make concepts like time travel inherently contradictory to our notion of computation? But the practicality of these exercises does not exceed their entertainment value.
- thinkharderdev 6y ago
  You are certainly correct that people arguing the opposite side probably don't have a formal system in mind, but I think the intuition that an open interval in the Reals doesn't have a smallest number is easy to grasp even without any formal training. So you can force them to see the consequences of it through fairly straightforward logical contradictions. Assume x is the smallest real number greater than 0. Then x/2 is also a real number and is greater than 0 but less than x. Therefore, x can't be the smallest real number greater than 0.

brlewis 6y ago

Why is 0.000... bigger than 0?

gspr 6y ago
It isn't.
- brlewis 6y ago
  You are correct. This was a rhetorical question to get traderjane to question whether "bigger than 0" really applies here.
- SkyBelow 6y ago
  Or is it? Say I'm a layman and I decide that in the system of math as I understand it, 0.000... is larger than 0. Yes, if I was going to be completely form with my own system of math I would eventually have to face the problems this introduces and resolve it, but until then I can generally adopt a self contradictory system and continue to live my life unaffected. Much like many people live their whole lives using naive set theory for their understanding of sets.
  - brlewis 6y ago
    Then in your system of math 0.999... is also less than 1. However, basic arithmetic taught to children requires that adding trailing zeros does not change the value of a number. You'll have a hard time doing arithmetic once you change that assumption.